A079201 -id:A079201 - cp's OEIS frontend

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448

Offset: 0

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Amit Sehgal, Sunil Kumar, Sarita, Yashpal, Commutative Associative Binary Operations on a Set with Five Elements, J. Phys.: Conf. Ser. (2018) 1000 012063
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.
Cf. A002489, A079192, A079195, A079198, A079201, A079210.

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).

a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

a(8) from Andrew Howroyd, Jan 26 2022

a(9) from Andrew Howroyd, Feb 14 2022

A079197 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 107, 0, 0, 0, 5, 0, 28, 488, 43389

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079194(n)+A079197(n)+A079200(n)+A079201(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,1; 1,4,5,107; 0,0,0,5,0,28,488,43389
A079195(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079196(x).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079194, A079198, A079199, A079200, A079201.

A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 4, 4, 0, 46, 73, 0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 84, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			Triangle T(n,k) begins:
  0;
  0;
  0, 0;
  0, 0, 4, 6;
  0, 0, 0, 4, 4, 0, 46, 73;
  0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079241.

Cf. A027423 (row lengths), A079175, A079201, A079202, A079203, A079204, A079205, A079197, A079208, A079209, A079240.

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079208(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079240(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k) - A079208(n,k). - Andrew Howroyd, Jan 27 2022

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 27 2022

A079208 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 0, 0;
  2, 0, 0, 0, 1, 0, 0, 0;
  2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums give A079243.

Cf. A027423 (row lengths), A079171, A079201, A079202, A079203, A079204, A079205, A079197, A079207, A079209, A079242.

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079207(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).

A079242(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079200 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 6, 2, 0, 0, 4, 5, 0, 46, 73, 2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 86, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Number of elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!).

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 4, 6;
  2, 0, 0, 4, 5, 0, 46, 73;
  2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets

Row sums give A079199.

Cf. A027423 (row lengths), A079171, A079194, A079175, A079197, A079198, A079199, A079201, A079210.

A079194(n,k) + A079197(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079198(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k). - Andrew Howroyd, Jan 26 2022

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079194 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 2, 0, 8, 66, 3115, 0, 1, 14, 18, 270, 467, 48260, 178888824

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			Triangle T(n,k) begins:
  0;
  0;
  2, 2;
  0, 8, 66, 3115;
  0, 1, 14, 18, 270, 467, 48260, 178888824;
  ...

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Row sums give A079193.

Cf. A027423 (row lengths), A079171, A079192, A079197, A079200, A079201, A079210.

T(n,k) + A079197(n,k) + A079200(n,k) + A079201(n,k) = A079171(n,k).

A079192(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=0 prepended by Andrew Howroyd, Jan 26 2022

A058105 Number of commutative asymmetric semigroups of order n.

Original entry on oeis.org

1, 1, 3, 9, 39, 201, 1258, 10013, 142755

Offset: 0

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Cf. A001426, A079201.

a(n) = A079201(n, A027423(n)).

Updated Feb 19 2001.
a(7) corrected Dec 26 2006
a(8) from Andrew Howroyd, Jan 26 2022

cp's OEIS Frontend

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

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A079197 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n, listed by class size.

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A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

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A079208 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

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A079200 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.

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A079194 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n, listed by class size.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A058105 Number of commutative asymmetric semigroups of order n.

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Crossrefs

Formula

Extensions